Commentary on Theories of Mathematics Education

Symbols and Mediation in Mathematics Education 229
Fig. 1 Goldin’s (2004)
schematic representation of
paths through the state-space
for the problem of the two
pails
written record, or perhaps are not sure how to do it—but without some systematic external
representation,thememoryloadishigh....Othersovercomethisimpassebykeepingtrack
systematically of the steps they have taken, or by persevering despite feelings of “getting
nowhere.” (p. 58)
On the other hand, there are cases when students begin to systematize their moves
into algorithms which when efficiently applied generate the result that one is after.
For instance in Sriraman (2004), ninth grade students (approx 14 years old) constructed
an algorithm to efficiently generate Steiner Triplets. Could this be viewed as
a natural pre-cursor to writing a computer program that efficiently generate triplets
for large start values of n. In fact many of them later went on to do exactly this.
Although Abramovich and Pieper (1996) recommend that teachers provide visual
representations (manipulative and computer generated) to illustrate combinatorial
concepts (arrangements, combinations etc.), an evolutionary perspective suggests
that it is better to have students generate the algorithm based on their scratch-work
(physical/mental) to produce the computer-generated representation. In a sense one
can view the process of generating an efficient algorithm to produce a computer generated
representation as the interface between the physical act of counting efficiently
(or systematizing moves), translating this efficiency into an algorithm (symbolism)
in order to get the computer generated representations.
Batanero and Godino (1998) analysis of the difficulties of children and adolescents
to fully conceptualize and understand the phenomenon of randomness (p. 122)
is compatible with the mathematician’s general view of probability theory as enumerative
combinatorial analysis applied to finite sets with considerable difficulties
to generalize the theory when considering infinite sets of possible outcomes.